the Earth and the Geological Changes of its Surface, 379 



^at^-^^ ; (5.) 



*. = -/*/„ '^'A4r'sinQ-jL^)cos^>!'*i*; (6.) 



When ^1'== -V and ^^ = — - > or when the above is inte- 



grated throughout the whole mass of the spheroid, it becomes 

 - 6912A r/25 „ \ 5 ^ ^ ,^ \,. ,^. 



When a becomes p and p becomes ^\ the density at tne 

 surface of the spheroid 



, A . 5 V. V rt;> ./. A = P f . 



p' = — sm - TT, - ~ .5 



V 6 sm— TT 



^ 6 



/, Substituting this value of A in (7.)> and remembering that 



8 

 ;Mi =-T7rp'Z»^^, Mj being the mass of the spheroid, we shall obtain 



- 864 r/25 o «\ ,5 , 2 5 1 <**,*- gibn 



I 



sm-x 



If the spheroid were homogeneous, we should have, after 

 making the necessary substitutions in (3.) or (6.), and inte- 

 grating throughout the whole mass of the body, 



^ / 2j^pr5cosWr(i*=:^i<.<2«''=|Maf,(9.) 



-fy 15 ■ 5 ,,^y 



a result easily obtained by the ordinary methods. 



In finding the moment of inertia of the external shell so as 

 to be able to appreciate the changes which it may undergo, 

 it should be remembered that as geological changes have not 

 the same magnitude or importance at every latitude, a change 

 in the moment of inertia of the whole shell can be found only 

 by considering the changes of the moments of inertia of its 

 different parts. biijo // noir^oitj 



Let the shell be supposed therefore tobonsist of a series of 

 zones, on each of which geological changes occur in a com- 

 paratively uniform manner, whatever may be the nature or 

 extent of the changes on any of the others. Let ^i, ^gf ^3» &c. 

 represent the moments of inertia of the zones in the northern 



